Preliminary Lemmas
6.3 Action on the Unitary Modules
T h en again th e proposition follows from Lemma 6.2.2.4, as we observe J >-*
( Ju J2) there defines a 1-1 correspondence from A ([n1—1]) to A ([n i—r —1]) x A ([r—1]) with | J | = l/il 4* |«/2| "b 1*
□
Next assume 712 > r. So «/U { r } \{ n2} — J U {r}. Let S be the set of subspaces of dimension r in V£- For R E S, let X ( R ) = { r E X \ (R2(t) Y = P } . Recall R i(r) = 0 for each r E X . By Rem ark 6.1.5.1, t P2CO) defines a G j-equivariant surjective m ap from X to {0} x {P* | R E S}. Therefore, as Nc r ( R ) = N c r ( R m),
X / G j = Ilife s /c r X ( R ) / { P r x * « * ) ) -
Let A C A (P ) be the set of flags in P (V?) of type J U {r}. For f l € S , let A (P ) be th e set of flags in A whose final term is R . T hen
A / G r = I L „ = s / c r A ( R ) / N a r ( R ) .
F ix R € S. Set P = Ng’-(R) an d P = A u tcr(R ). As R i ( r ) = 0 for all r E X , Cqx(Vi/ Ri(t)) = 0 and A u tp+ r(V i/R 1(r)) = P j r. By Proposition 6.1.4.5, X (R) / ( P j r x P) is in 1-1 correspondence w ith P \G +r/ P j r and th e stabilizer in G j of th e orbit corresponding to P g P j r is th e extension of C p ( R) by P g fl P j T.
O n th e other hand, there is a n P-isom orphism between A(R) and th e set of flags of type J in V { R ) given by c (->• c\{i2}, where V { R ) is the poset of proper subspaces of R.
As G +r is transitive on th e chains of type J in V { R ) and P j r is the stabilizer in G +T of a chain of type J , it follows th a t A( R ) / P = A( R ) / P is in 1-1 correspondence w ith the P -o rb its on G+r/ P j r, or equivalently P j r\ G +r/P , such th a t th e stabilizer in Gr of the o rb it corresponding to P j rg~l P is th e extension of Cp(R) by P f l ( P j r)9 l - We have established a 1-1 correspondence between X ( R ) / ( P j r x Ng^{R)) and A( R) / Nc' - ( R) for each R E S /G r with th e desired property. So the lemma follows.
□
D e f in itio n 6 .3 .2 . Assume = r and J C [ni — 1]. If tGj E Irr(V, r ) / G j corre
sponds to E A(P(V^*)) of ty p e J U { r } \{ n 2} as in Lemma 6.3.1, we say r Gj is labeled by c6^ . By abuse of notation, we may also say r is labeled by c and write
T = Tc .
Let A = A(P(K>*)), and T th e set of norm al chains in V ( y 2 ) as defined in section 5.2. Recall from the paragraph preceeding Proposition 5.2.17 th a t A (r), T (r) are th e r - th truncations of A, T, respectively, an d in particular A(ri2) = A , r ( r i2) = T.
L e m m a 6 .3 .3 . A ssum e ni = r . Let E be an abelian group and f : V x X -+ E a G-stable function o n V x X .
(1) There is a 1-1 correspondence <f>: (V x X ) f G —¥ A ( r ) //? 2, such that if (cj, x )G •-» cP2,
then d is o f type J U { r} \{ n 2} and GCj,x = N n2(c).
(2) Let g be the 122-stable function on A defined by g = f o 0_1. Assum e g can be extended to an 122-stable function in the sense o f Rem ark 5.3.3. I f r = n 2, then
A { T { f ) , V / G ) = A(g, A/122) = A(g, T /122).
I f r < n 2, then
A { T { f ) , V / G ) = -A (^ , A (r)/A > ) = - A ( ^ , r ( r ) / / 2 2).
Proof. Recall from Exam ple 3.1.1 th a t c j is th e flag stabilized by P j r, and {cj- J C [m - 1]}
is a set of representatives of V /!2 \. So
( V x X ) / G = U ({cj } x X ) K P ? x n 2) < z x / G j .
J C [r - 1 ]
Here = means 1-1 correspondence. W ithout loss we m ay identify these sets.
Set J = J U { r} \{ n 2}. By Lemma 6.3.1, there is an 1 - 1 correspondence
<t>j '• X / G j —>• A j(r)/1? 2
x Gj i-f (f>j(x)n*
where A j ( r) denotes th e set of chains of type J in A (r), such th a t GCjiX = (<f>j(x)).
Therefore, as A (r) is th e disjoint union of A j , <f) = U j<f>j w ith (f>((cj, x ) Gj) = <pj(x)a2 is the desired 1-1 correspondence. P a rt(l) holds. As the value of each i?2-orbit on.
A (r) under g is determined by / and 0, g is a well defined ^ - s t a b l e function. We have
A ( r ( / ) , p / G ) = Y ( - i ) |,|/ ( c , i ) = ± E
(c,i)e(A(P)xX)/G deA/fia
= ± A (0, A /f?2)
where th e + is taken if r = 712, and — is taken if r < 712. This is because when (c, x) corresponds to d under <f> with c of type J , th e d is of type J. B ut J = J if r = n 2 and J = J U {r} if r < n 2.
This proves th e first equality of p a rt (2). T h e second equality follows from Remark 5.3.3 and Proposition 5.2.16. So th e proof is complete.
□
For each J C [m — 1], we denote by S U(V\ J , r) C X / G j th e set of r G X / G j labeled by norm al chains of type J U { r } \{ n 2} in V(V{).
L em m a 6 .3 .4 . Assume ni = r . Let d ^ 0, Z ^ Z( G) centralizing V and p £ Itt( Z ) . Then
E (-1
)'J' k 4 H j , X , p ) =Y
J C [ r - l ] JC [i— 1]
Proof. We define a G-stable function / on V x X as follows. For J C [r — 1] and
t £ X f H j , let / ( c j , r ) = kd{Hj, r ,p ) . As in th e proof of Proposition 6.2.1, this is a well defined G-stable function on V x X w ith T ( /) ( c j) = kd(H j, X , p), so th a t
A ( T ( f ) , P / G ) = Y ( - 1
J C [ r -1]
Let 0 and g = f o f>~1 be as in Lemma 6.3.3.
If c £ T (r) is of type J as in th e preceeding lemma, then g(c) = kd{Hj, r, p) where 0 ( c j , r ) G = cP2. Consequently as <j> is a 1 - 1 correspondence, and S U(V, J, r) is in 1-1
correspondence w ith the T (r)/i22,
A ( a ,V /n - l) = ± J 2 ( . - l ) 'J' h ( K j , S “( V ,J ,r ) ,p ) .
J Q [ r -1]
O n th e o th er hand, as Hj splits over V w ith V abelian, so by Lemma 2.2.3 and 2.2.4, g(c)
=
kd-di (Nn2(c), p) where d\ is th e exponent of q inI
Gj/ Ng{c j,t)\= |
G/Nn2(c)\by Lem ma 6.3.3.1. Therefore, if Gc
=
G v, th en g(c) = g(d). So by Remaxk 5.3.3, g can be extended to a i?2-stable function. Therefore th e lemma follows from Lemma 6.3.3.□
We now discuss th e case when ni > r . Pick w € Y as in Proposition 6.2.1. Let V — (Vi/w)® V2. Recall dim(Vi/w) = r an d A utfi^V i/u;) = G+r. Then V is a tensor m odule for G+r x Q2. So the above discussion applies to X = Irr(V', r).
Form the semi-direct product H = V x (G+r x S?2), and let Hj2 = V x Gj2,
Gj 2 = P jJ x f22 for J2C [r — 1]. By Lemma 6.3.1, X / Hj2 is labeled by G n2-orbits of chains of type J 2 U { r} \{ n2} in V i V f ) .
L e m m a 6 .3 .5 . Assume r < n\. There is a Aut^ (Vi/w) x Q2-equivariant isomor
phism 6 between X(w) and X given by 6(r)
=
r, such that for r 6 J C \nx —1] and r 6 X(w),NgA t) = X W p jf X O .M =
where J\ = { j —r \ j € J{> r)} and J2= J{< r ) .
Proof. By Lem ma 6.1.7, A ut«1(Vri/iw) = G+r. T he existence of Qfollows from R em ark 6.I.5.2. L et J and r be as in th e hypothesis. By Lemma 6.2.2.3,
Np r ,(w) = x p*;
w ith
C p ^ iV i/w )
=
P ^ ni~r) an d A u t ^ (Vt /w ) = P + r .By Proposition 6.1.4.5, C p + ^fV i/w ) ^ Nq j( r ) . As C p ^ ( V i / w ) commutes with the diagonal subgroup D = A utp +-H (V i/w ) (described in the proof of Lemma 6.2.2) and f?2, and as Ngj ( t) ^ Np-^i (w) x i?2, we have
x N p};>clh(r).
Now as Gj2 — P j 2r x j? 2 and 9 is A u t n ^ V i / w ) x /22-equivariant, it follows th a t
Therefore, th e lemma is proved.
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D e fin itio n 6 .3 .6 . Let r < n i and 9 be as in Lemma 6.3.5. For r E «/ C [ni — 1], we say t Ngj^ E X ( w ) / N g j ( w ) is labeled by <P2 E P (V 2*)fGr of type J2 LI { r} \{ n 2} if
9 ( t ) G j 2 is. By abuse of notation, we also say r is labeled by c and w rite r = rc.
Let r < T i i• We denote by S U(V, J , r) C X ( w ) / N g j(w ) th e set of orbits labeled by a norm al chain of type J2 U { r} \{ n 2} in P 0 4 * ).
R e m a r k 6 .3 .7 . In general, each x E S U(V, J, r) is not a member of X / G j , but x Q y for some y E X / G j . Hence for r E x, by Lemma 2.2.1.2,
kd{Hj, x, p) = kd{ H j, r , p) = kd( H j, y, p).
So for this purpose, we may regard S U(V, J , r) as a subset of X / G j .
L e m m a 6 .3 .8 . Assume r < n i. Let d, Z and p be as in Lemma 6.3.1. Then Y ( - 1 )W U B j , X , p ) = Y { - T ) Wkd( H j , S “( y j , r ) , p ) . (6.7)
jc[m -l] r€JC[m-i]
Proof. By Proposition 6.2.1, we have
Y ( - 1 Y I" 1 )[AU H j , X { w ) , p ) (6.8)
jc[m -i] reJC[m-i]
where w is defined in th at proposition. Pick r E X ( w ) . By Lemma 2.2.2,
kd(Hj, r, p) — kd_d, ( N Hj( r ) , r, p)
where d! is the exponent of q in th e p -p art of \Hj/ N Hj(t)\. B ut H j = V x G j with V abelian, so by Lemma 2.2.4,
kd-df (NHj ( r ), r , p) = kd- d> (NGj ( r ) , r , p) .
Ng j(t) is given by Lemma 6.3.5. So
k A H j , r <P) = kd^ r ( P g ' " - T) x N 0 j t (r),p)
where J\, J2 are as in th at lemma. Similarly as Pj^ni~r^acts trivially on X and Hj2= V x Gj2with V abelian, by Lem m a 2.2.2 and 2.2.4,
ki - * + r ( P * '" ~ T) x = ki - * ( P X l" ' - r) x N 0j, ( f ) , p ) where d" is the exponent of q in th e p -p art of \HjJ N r (f)\. Therefore,
kd{Hj, r, p) = kd- d'+d»( p + (ni_r) x H j2 , f , p ) . (6.9) Recall H j = V x G j with V ^ Nh3 {t) and G j = P j ni x Q2. So
\Hj\ / \ N Hj{t)\ = \Gj\/\Ng j(t)\.
Similarly Hj2 = V * Gj2 with V ^ N Rj2(r) and Gj2 = P j (ril_r) x Q2. So
\ Sj\/\N Rj(t)\ = \G j\/\N G j(f)\.
B ut by Lemma 6.3.5,
V a A r ) = x N Sj3(f).
Therefore, d' — d" is the exponent of q in th e p -part of \ P j ni/ ( P j ^ ni~r^x P j ^ ) \ , which is 2((T£) — (ni^"r) — Q ) , and does not depend on the choice of J.
E quality (6.9) holds for any r G X ( w ) , so as J, r) (resp. X ) is in 1-1 correspondence w ith S U(V, J2,r) (resp. X ) ,
k i( H j , S“ (V, J,r), p) = X H j ' , S “( y , J2, r) ,p ) and
kd( H j , X ( w ) , p ) = kd- d'+d'.(PZ<-ni- r) x H j 2, X , p ) .
E
( - 1 ),M k d ( H j „ X , p ) = Y , ( - 1-/2 C [ r - l ] / 2 C [ r - l ]
Applying Lemma 2.2.6, we obtain th a t for any d ^ 0,
E E (-i),JiM J:,{kd(pxl'"^) *Rj„x,p) =
J i C [ m - r - l ] J-2 C [ r - l ]
E E (-1 yJiW*kAp£’n~r) *R.h,s'{v,j*,r),i>).
/ i C f m - r - l ] J-2C [ r - l ]
Recall th a t J ( J i, J2) defines a 1-1 correspondence between th e subsets of [nL containing r and A ([ni — r — 1]) x A ([r — 1]) w ith | J\ = | J \ + \J2\ 4-1. So
Y ( " I )'J'k d ( H j,X ( w ) ,p ) =
r G / C [ n i —1]
- E E (-1 )l*“ lw ( f ? ,H xH4,xp).
J i C [ n i - r —1] J 2 C [r—1]
Similarly
Y ( ~ l ) ' J'kd(Hj, S*(V, J,
r),
p) =r € / C [ n t —1]
- E E (-l)W*W W (fJl“',) x
• /iC [ n i—r —1] —1]
By (6.10), the last two double sums are equal, so
r e J C [ m - l ] r e / C [ n i —L]
Hence th e lemma follows from (6.8).
Fix J C [tl-i — 1] and let w = 0 (if r = n i) or w be as in Proposition 6.2.1. We have labeled X ( w ) / N ’g j (w) by Gr-orbits of chains of type J U { r} \{ n 2} in V iV ^ ) when r = n i or r 6 / , or equivalently when r E J U {7 1 1}- In particular S U(V, J, r) is defined when r € J U {ni} and in 1-1 correspondence w ith th e G712-orbits on the norm al chains in V iV ^ ) of type J U { r} \{ n 2}, in which case S “ (V, J, r) is non-empty as norm al chains of any type exists in V iV ^)- Set S U(V, J, r ) = 0 i f r f £ « / U {nL}, and set
m in (n i,ri2)
s ‘ < y , J ) = 1 J s “( v , j , r ) .
r = l
Furthermore, let S^iV , J, 0) consist of the trivial character of V .
Recall th a t Irr1(Vr) is th e disjoint union of Irr(Vi r ) for 1 ^ r ^ m in(n1, n 2) and each Irr(V, r) is a G-set. So it follows from Lemma 6.3.4 an d 6.3.8 by summing over all r w ith 1 ^ r ^ m in(ni, n 2) th a t
P r o p o s itio n 6 .3 .9 .
( - l ) '- 'll £ ( H j , V , p ) = J 2 ( - 1 (6.11)
7C[m-i] yc[m-i]
Recall the definition of singular c h a in s as well as non-singular chains from Defini
tion 5.2.14.
D e fin itio n 6 .3 .1 0 . Let J C [m — 1], 1 ^ r ^ m in(ni, 722) and r € J with J = J{< r) U { r} \{ n 2}.
(1) Assume r < n 2; so J = r). Let S SU(V, J , r ) be th e set of r 6 S u( V , J , r) labeled by a singular normal chain of type J ( ^ r) in V (V £ ). Observe S 3U(V, J, r) is non-em pty if and only if r ^ n 2/2 , in which case it consists of a unique member. Let S nu(V, J, r) be the set of r € S U{V, J, r) labeled by a non-singular normal chain in P (V ^). Clearly
S U(V, J ,r) = 5 SU(V, J ,r ) .
For 1 ^ r' ^ r , let S£U(V, J, r) be the set of members in S nu( V ,J ,r ) labeled by non-singular norm al chains of non-singular ran k r'. Observe S ^U(V, J, r) is non-em pty if and only if r1 G J = r) and J {< r') C [r'//2]. Clearly
r
7^=1
(2) Assume r = n 2; so J = / ( < r). For 1 ^ r 7 < r , let 5 " U(V, J, r ) be the set of members in S u( V , J , r) labeled by non-singular norm al chains o f type J (< r) in V { y £ ) non-singular rank r'. Observe S™(V, J, r ) is non-em pty if and only if r ' E J = / ( < r). Let S™(V, J, r) be the set of members in S U(V, J, r) labeled by singular normal chains of type J = J ( < r ) in V iV ^ ) . Observe S?(V, J, r) is non-em pty if and only if J{< r) C [r/2]. Set
r
S ™ (V ,J,r) = ] \ S ? ( V , J , r ) .
r'=l
So in this case S U(V, J, r) = S nu(V, J, r), and we m ay set S 3U(V, J, r) = 0.
(3) Set
m in ( n i ,n 2 )
sn(v,n= (j
r=l
m in ( n x ,n 2 )
s?(v,r>= IJ s?<y,j,r),
r=r'
and
m i n ( n i ,r i2)
S nu(V ,J) =
U
^ ( V , J,r).r=l
R e m a r k 6 .3 .1 1 . Observe in either case of Definition 6.3.10, S ^ V , <7, r) is non
em pty if and only if 1 ^ r < 712/2, in which case it consists of a unique member.
S™(V, J ,r) is non-em pty if a n d only if r' G J { ^ r) an d J ( < rf) C [r^/2]. Moreover, S U(V, J, r) = 5 “ (V, J, r) Q S ~ ( Y , J, r)
and
S U(V, J) = S au(V, J) U S nu(V, J).
We make the following observations.
R e m a r k 6 .3 .1 2 . Let J C [m — 1] with r G J U {n i}.
(1) Assume r G 5 “ (V, «/, r ) . If r < rci, by Lemma 6.3.5, JVcJ ( r ) = F i (’" - r ) x i V ^ ( f ) .
Here J i = {j —r | j G J { > r)} and J i = J{< r). If r = nj., ro as in Lemma 6.3.5 may be chosen to be 0, so th a t H , V , H j 2 , f are identified w ith H , V , H j, r , respectively. So the above equation holds trivially in this case as pjKn i-r) = i and N g j ( t ) = N q J2 (r). In either case, f is labeled by a norm al chain c in P(V ^) of type J = J2U { r } \{ n 2}- Recall from the discussion preceeding Lemma 6.3.5 th a t f G l r r ( y , r ) w ith V = Mnn2(F,a). So we m ay apply Lemma 6.3.1 to get
nGj2{t) = l V G n2 ( c ) .
Consequently
N Gj(t) = P * ^ x N c n ( c ) .
(2) Assume r = rc G S 3U(V, J, r) is labeled by a singular norm al chain c in V iV ^), then N n2(c) — P j a. Therefore,
NgAt) = P j,<" '" r> * P ? - (6-12)
(3) Assume r = rc € 5J5“ (V, J, r). Then c is a non-singular norm al chain of type J in V ( V f ) of non-singular rank r'. So 1 ^ r7 ^ r an d J { < r') C [r'/2]. Let c correspond to (ci, C2) as in Lemma 5.2.15.3, where Ci is a singular normal chain in 'P (V T') and c2 is a norm al chain in V ( V n2~r'). Here V r' is th e n atu ral module for G r' and V n2~T' is sim ilarly defined. Ci is of type J ' = «/(< r') = J{< r’) and c2 is of type J " = { j — r ' \ j e J2{> r ')} w ith \ J\ = | J'\ + | J"\ 4 -1. So by Lem m a 5.2.15.3 and by p a rt (1),
NcAt)
= x JVCci) x
w ith N Gt'{c\) = P j\<r>) being a parabolic subgroup of . P r o p o s it i o n 6 .3 .1 3 . (1) I f r = n i, then
( - 1 )'J'kd( H j , S “ (V, J, r ) , p ) =
J Q [m -1]
0,
X 3 j C [ n 2/2 ] ( ~ ^ ) k d - d ' { P j u { r } > P ) >
^ I f r < n i, then
J 2 ( - 1) l J [ k d ( H j , S au(V, J, r), p) =
J C [n x- l ] /
0, i f r = n 2;
• E A C [« /2| ( - l ) |-,,l+IAI+l X
i f r < n 2.
k.
/ n any case d! = 2((”1) — (ni2~r) ) -
Proof. If r = n 2, S'stt(V, J, r) = 0 by definition. So th e statem en t holds in this case.
W ith o u t loss assume r < n 2.
F irst assume r = n i. From Rem ark 6.3.11, S ^ V , J ,r ) consists of a unique m em ber if J C [r/2], and is em pty otherwise, in which case kd (H j, S 3U(V, J, r), p) = 0.
i f r = n 2;
i f r < n 2.
For r E S SU(V, J ,r) (and hence J C [r/2]), from Rem ark 6.3.12.2, N Gj(r) = P ”2. So as H j is th e semi-direct product of an abelian norm al subgroup V by G j, by Lemma 2.2.1.2, 2.2.3 and 2.2.4, we have
kd( H j,S ™ ( V ,J ,r ),p ) = kd( H j , r , p ) = k d_d,(N Gj(T),p) = k d- d, ( P ? , p )
where d! is the exponent of q in \G j \/ \ N G j(t)[. AsG j = P j n i x.Gn 2 and N G j( t) = P” 2
and the P j 2 has the same g-height as G"2, it follows th at d' = 2(»*).
Therefore, the part (1) holds. T he case r < n x can be proved similarly.
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To end this section, we prove th e following technical lemma which will be used in section 9.2.
Let J C [ni - 1], 1 ^ r 6 J U {n i} T ^ n2• Let J = J ( < r) U {7~}\{n2} and r' E J- Let V = Mni_r/i„2_r/(F) be a tensor module for G = G+^ni~r'^ x Gn2-r/. Form th e semi-direct product H of V by G, and for J ' C \nx — p — 1], set Hj> = VG j>
where Gj> = P j (rn_r,) x Gn2-r/.
L e m m a 6 .3 .1 4 . There is a 1-1 correspondence 7 fro m S™(V, J, r ) to S^^V , J r —r') where J ’ = { j — P \ P < j E J } , such that f o r r E S™(V, J, r),
NoJ t) = x N BjA7 W ) .
Proof. By definition, there is a 1-1 correspondence a v j from J, r) to the set A of Gr-orbits on the nonsingular norm al chains of nonsingular rank r7 in P (V n2) of type J . Moreover, by Remark 6.3.12.1, if r = r c, then
NgAt) = X N a ^ (c) (6.13)
where J x = { j — r \ r < j E J } . By Lemma 5.2.15.3, and as GT> is transitive on th e set of singular normal chains of a given type in P ( Y r'), there is a natural 1-1 correspondence 0 from A to the set B of Gn2_r,-orbits on the set of norm al chains in V { V n2~r) of type
J ' = { j — r' | r' < j G J } = J r U { r — r'} U { n 2 — r'}, such th a t if c G A w ith d = 9(c), then
Ng* 2( c ) = - P / ( < r /) x N Gn2-Tj ( c ) . ( 6 - 1 4 )
Finally by j, is a 1-1 correspondence from S U(V, J', r — r') to B such th a t if r7 = r^ G
S U(V , J', r — r1), th en
with
= { / - ( r - r1) | f G J '( > r - r')} = {(j - r') - (r - d ) | j G J } = J i.
T h a t is,
= •P,+(" ‘- ) x A U - ' M (6-15) Now it is easy to check th a t 7 = avtjo 9 o b p lJf is a 1-1 correspondence from S*U(V, J, r) to S ^ i y , J', r — r'). If r r imder this map, then by equations (6.13)-(6.15),
WG j(r) = P X ^ x x = P j; < 0 x Ar< v (r').
Therefore, th e proof is complete. □